Coherent Control in Nanolithography: Rydberg Atoms†
Nam A. Nguyen, Bijoy K. Dey,‡ Moshe Shapiro,§ and Paul Brumer* ,|
Chemical Physics Theory Group, Department of Chemistry, UniVersity of Toronto,Toronto, Ontario, M5S 3H6, Canada
ReceiVed: January 20, 2004; In Final Form: April 27, 2004
A technique based on coherent control for the optical manipulation of deposition patterns in nanofabricationwith neutral atomic beams is described. The theory, optical nanolithography with bichromatic fields, is thenapplied to the deposition of rubidium Rydberg atoms on surfaces. Controllable nonperiodic patterns anddeposition lines are shown to be possible due to interference contributions to the induced polarizability.
I. Introduction
Traditional photolithography, the mainstream technique forfabricating integral circuits in the microelectronics industry,comprises two steps: patterning of a resist mask and etchingof the unmasked regions. During the past three decades the sizeof integral circuits has decreased considerably, from themicrometer regime to∼100 nm obtained routinely today. Thisrate of miniaturization is, however, expected to slow when theinherent physical limitations due to atomic size of the substrateand the diffraction limits of the light sources are reached.
Currently, several alternative techniques are being proposedand developed.1 One possible approach uses neutral atoms ormolecules as a material source and coherent light as a focusinglens;2-10 i.e., the roles of the light source and the mask arereversed with respect to the traditional photolithographic process.This technique has been made possible by recent advances inlaser technology. In contrast with the traditional two-stepapproach, it is direct write; that is, the need to remove theunmasked region is eliminated, causing less damage to thesurface.
Manipulating matter beams with light relies on the fact thatlight carries momentum, so that a slowly moving atom, incollision with a photon, gets deflected and/or focused. Thismakes possible the creation of various atom-optical elements,such as lenses,11-13 lens arrays,14,15 mirrors,16-19 beamsplitters,20-22 and waveguides.23,24There are two types of forcesat work: radiation pressure and the optical dipole force. Thelatter, acting as confining mechanism, was first considered byAskar’yan26 in connection with plasmas and neutral atoms. Thepossibility of trapping atoms with this force was then exploredby Letokhov27 who suggested that atoms might be one-dimensionally confined at the nodes or antinodes of a standingwave tuned above (blue-detuned) or below (red-detuned) theatomic transition frequency. Ashkin28 demonstrated the trappingof micron-sized particles in a laser light based on the action ofthe dipole force and later suggested29 a scheme for three-
dimensional neutral atom traps. The dipole force on neutralatoms was also demonstrated by Bjorkholm et al.11 by focusingan atomic beam using laser light. Chu et al.30 exploited thisforce to realize the first optical trap for neutral atoms. In theearly 1990s optical dipole forces began to attract increasinginterest (see refs 31 and 32), not only for atom trapping butalso in the emerging field of atom optics, including nanolithog-raphy. Recently, efforts have been made to extend nanolithig-raphy to include molecules.33-35
In atomic nanofabrication, the topic of this paper, one affectsthe center of mass motion using the dipole interaction betweenthe atoms and a spatially nonuniform electric field, leading tothe confinement of the atoms in regions of space whose size isonly a fraction of the optical wavelength. A periodic array ofsuch dipole force causes an atomic beam to form a periodicpattern on a substrate. The advantage of atomic nanofabricationis that neutral atomic beams are simple and inexpensive sourcesof particles with de Broglie wavelengths< 1 Å. The trajectoriesof neutral atoms are unaffected by uniform electric or magneticfields, and the long-range interparticle forces between neutralatoms are small. Also, neutral atoms have laser-accessibleinternal structures that permit laser cooling,25 to enhance theflux and collimation of atomic beams.
Almost all of the work done thus far in optically inducedatomic nanolithography has produced deposition patterns con-sisting of periodically repeated parallel lines or an array ofordered atomic dots on a substrate.2-10 Recently,41 we suggesteda way of overcoming this limitation in order to producecontrolled nonperiodic patterns by using the bichromatic coher-ent control scenario. In general, coherent control affects theoutcome of a chemical and physical events by manipulatingquantum interferences between different excitation pathways thatlead to the same state (for reviews, see refs 37-39). Inparticular, using such interferences has proven effective incontrolling molecular (and atomic) polarizabilities and, byextension, refractive indices,40 resulting also in the control ofnanoscale deposition patterns obtained when molecules (suchas N2
41) traverse strong electromagnetic fields. In this paper wefurther explore the ability to produce aperiodic patterns whosestructure can be manipulated by altering the parameters of theincident electric fields. Further, we do so with Rydberg atoms,whose large polarizability allows the use of relatively lowintensity fields.
The paper is organized as follows. Section II presents thetheory behind the coherent control of atomic and molecular
† Part of the special issue “Richard Bersohn Memorial Issue”.* Corresponding author.‡ Current Address: Department of Chemistry, McMaster University,
Hamilton, Ontario, L8S 4M1, Canada.§ Permanent address: Department of Chemical Physics, The Weizmann
Institute of Science, Rehovot, 76100, Israel, and Departments of Chemistryand Physics, University of British Columbia, Vancouver, V6T 1Z1, Canada.
| Also, Center for Quantum Information and Quantum Control, Universityof Toronto.
dipolar light-induced potentials using bichromatic fields. SectionIII provides the computational methodology used in calculatingthe light-induced potential and the trajectories of atoms travers-ing multicolored electromagnetic fields. Results of the depositionpatterns for rubidium atoms in intermediate and high Rydbergstates thus obtained are presented in section IV. Conclusionsare drawn in section V.
II. Atomic Deposition via Bichromatic Control
Consider an atomic system in a superposition state interactingwith a bichromatic electromagnetic field that runs parallel to asurface upon which the atoms will be deposited. The entiresystem is governed by the light-plus-matter Hamiltonian, givenin the dipole approximation by
wherer cm is the atomic center-of-mass coordinate with associ-ated kinetic energy operator,K(rcm). The vectorrel is the internal(for atoms this is the electronic) coordinate, withHel(rel)denoting the matter Hamiltonian. We denoter ≡ {rel,r cm} with- µ(r )‚E(r cm, t) as the light-matter interaction in the dipoleapproximation. The termµ(r ) represents the electric-dipoleoperator, andE(r cm, t) is the electric field at the atom center-of-mass.
With x denoting a coordinate parallel to the deposition surface,we assume a bichromatic standing wave with electric field ofthe form
where each term is the sum of two counter-propagating CWfields:
whereηi denotes the polarization direction of theith field. Herethe two polarizations are taken to be parallel to one anotherand perpendicular tox, and aligned, as depicted in Figure 1,along the laboratory framez axis which is taken to beperpendicular to the surface. We denote the spatial part of theith standing wave field byFi(x) ≡ cos(kix + φi), and the relativephase between the two fields byφF ≡ φ2 - φ1.
We choose the initial atomic state to be a superposition state:
where Φi and pωi are, respectively, two eigenfunctions andeigenvalues ofHel(rel),
Given eq 4, the frequencies of the CW fields are chosen suchthat
Basically, the light field affects the trajectories of the atomiccenter-of-mass motion throughVLIP, the “light-induced poten-tial”, which results from the interaction of the electric field withan atomic dipole moment⟨µind⟩ that it induces. That is,
Altering this potential (through interference or other means)allows for the manipulation of the resultant deposition pattern.
A. Controlling the Light-Induced Dipole. We can separatethe center-of-mass motion from the much faster electronicmotion using the adiabatic Born-Oppenheimer approximationto obtain a Schro¨dinger equation in the space of the electronsthat is parametrically dependent onr cm:
For weak fields, first-order time-dependent perturbation theorycan be used, according to which the solution of eq 8 is givenby
The subscript e,e > 2, refers to the excited states ofHel(rel)with respect to the initial superposition stateΨs(t). Below, weare concerned with center-of-mass motion along one direction.Hence, we replacer cm by the coordinatex. The coefficientsce(t) in eq 9 are given by
whereµe,i(k) ) ⟨Φe|ηk‚µ|Φi⟩ are the transition dipole moments.
To avoid divergences on resonance, i.e., whenωe,i ) ωkF, we
Figure 1. Schematic representation of the lithographic experiment forthe Rb atom beam in a Rydberg state. The schematic on the left-handside shows periodic deposition from a single Rydberg state. Theschematic on the right-hand side shows controlled deposition from asuperposition of Rydberg states.
ip∂
∂tΨel(rel, t|r cm) )
[Hel(rel) - µ(rel|r cm)‚E(r cm, t)]Ψel(rel, t|r cm) (8)
Ψel(rel, t|r cm) ) Ψs(t) + ∑e>2
ce(t)Φe(rel) exp(-iωet) (9)
ce(t) )1
p∑i)1
2
bi∑k)1
2
µe,i(k) Ek
(0)Fk(x)/
2{exp[i(ωe,i - ωkF)t] - 1
ωe,i - ωkF
+exp[i(ωe,i + ωk
F)t] - 1
ωe,i + ωkF } (10)
H ) K(r cm) + Hel(rel) - µ(r )‚E(r cm, t) (1)
E(x, t) ) E1(x, t) + E2(x, t) (2)
Ei(x, t) ) ηi
Ei(0)
2cos(kix + φi)[exp(-iωi
Ft) + c.c.] i ) 1, 2(3)
Ψs(t) ) b1Φ1 exp(-iω1t) + b2Φ2 exp(-iω2t) (4)
Hel(rel)Φi ) pωiΦi, i ) 1, 2 (5)
ω1F - ω2
F ≡ ω1,2F ≈ ω2,1 ≡ ω2 - ω1 (6)
VLIP ) - ⟨µind⟩‚E(r cm, t) (7)
Coherent Control in Nanolithography J. Phys. Chem. A, Vol. 108, No. 39, 20047879
add below to the denominators of the final expression factorsof - iΓe/2 whereΓe are experimentally determined line widths.
It follows from eq 10 that the probability|ce(t)|2 of excitationfrom the initial superposition stateΨs(t) to a stateΦe arisesthrough three different terms: (1) the excitation ofΦe, havingstarted fromΦ1, with probability proportional to|b1|2; (2) theexcitation of Φe, having started fromΦ2, with probabilityproportional to|b2|2; and (3) the interference between these twopathways, which is proportional to 2Re{b1
/b2} ) 2|b1b2| cos-(φM), where φM is the (experimentally controllable) relativephase betweenb1 andb2. Note that this interference term owesits existence to the presence of both a bichromatic field and acoherent superposition state, to which the two colors of the fieldare coupled.
Using eq 9 we can write an expression for the expectationvalue of the dipole moment:
where “c.c.” denotes the complex conjugate of the term thatprecedes it.
The dipole moment is seen to be a sum of a “fieldindependent” and a “field induced” terms
where the field independent term is given by
The field-induced term⟨µind(x, t)⟩ depends once(t) [eq 10] and,after neglecting the quadratic terms in accord with the weakfield assumption, is given by
Thus, the induced dipole moment can be controlled by properlychoosing φF, φM, or |b1/b2|. Below we show that theseparameters are useful in controlling the deposition pattern on asurface. In doing so we focus, as a specific example, on thecase of hydrogenic-like wave functions⟨rel|n,l,m⟩, wheren, l,m denote the principal, angular momentum, and magneticquantum numbers, respectively. In particular, ifΦ1 is describedby then1, l1, m1 quantum numbers, then we choosen2, l2, m2 ofΦ2 asn2 ) n1, l2 ) l1 + 2, andm2 ) m1.
B. The Light-Induced Potential. The translational motionof the center-of-mass of the atoms is governed by the spatiallyinhomogeneous induced dipole which gives rise to a “light-induced” potentialVLIP(x, t). This potential, given by
has two components,
whereV(1)(x, t) is linear in the electric field and arises from thefield-independent dipole of eq 13, andV(2)(x, t) is quadratic inthe field and arises from the field-induced dipole of eq 14. Thatis,
Inserting eq 14 into eq 17 gives the quadratic part of the light-induced potential as
The potentialV(2)(x, t) can be written as a sum of two terms
whereVnon(x, t), the non-interference term, is obtained from thei ) i′ part of V(2)(x, t) of eq 19 andVint(x, t), the interferenceterm, is a result of thei * i′ contribution toV(2)(x, t).
Consider then these two contributions. The first is given by
VLIP(x, t) ) V(1)(x, t) + V(2)(x, t) (16)
V(1)(x, t) ) - ⟨µz(t)⟩ ∑l
El(x, t)
V(2)(x, t) ) - ⟨µind(x, t)⟩ ∑l
El(x, t) (17)
V(2)(x, t) )-1
4p∑k)1
2
∑l)1
2
Ek(0)Fk(x)El
(0)Fl(x) ∑i)1
2
∑i′)1
2
∑neleme
bi′/biµi′,e
(l) µe,i(k)
{exp[i(ωi′,i - ωkF - ωl
F)t] - exp[i(ωi′,e - ωlF)t]
ωe,i - ωkF
+
exp[i(ωi′,i + ωk,lF )t] - exp[i(ωi′,e - ωl
F)t]
ωe,i + ωkF
+
exp[i(ωi′,i - ωk,lF )t] - exp[i(ωi′,e + ωl
F)t]
ωe,i - ωkF
+
exp[i(ωi′,i + ωkF + ωl
F)t] - exp[i(ωi′,e + ωlF)t]
ωe,i + ωkF } + c.c. (18)
V(2)(x, t) ) Vnon(x, t) + Vint(x, t) (19)
Vnon(x, t) )-1
4p∑k)1
2
∑l)1
2
Ek(0)Fk(x)El
(0)Fl(x) ∑i)1
2
∑neleme
|bi|2µi,e(l) µe,i
(k)
{exp[-i(ωkF + ωl
F)t] - exp[i(ωi,e - ωlF)t]
ωe,i - ωkF
+
exp[iωk,lF t] - exp[i(ωi,e - ωl
F)t]
ωe,i + ωkF
+
exp[-iωk,lF t] - exp[i(ωi,e + ωl
F)t]
ωe,i - ωkF
+
exp[i(ωkF + ωl
F)t] - exp[i(ωi,e + ωlF)t]
ωe,i + ωkF } + c.c. (20)
⟨µ⟩ ) b2/b1µ2,1 exp(iω2,1t) + c.c.+
∑i′)1
2
∑e>2
ce(t)bi′/µi′,e exp(iωi′,et) + c.c.+
∑e′>2
∑e>2
ce′/ (t)ce(t)µe′,e exp(iωe′,et) (11)
⟨µ⟩ ) ⟨Ψel|µ|Ψel⟩ ) ⟨µI(t)⟩ + ⟨µind(x, t)⟩ (12)
⟨µI(t)⟩ ) b2/b1µ2,1 exp(iω2,1t) + c.c. (13)
⟨µind(x, t)⟩ )1
2p∑i′)1
2
∑i)1
2
∑k)1
2
Ek(0)Fk(x)
∑e>2
bibi′/µi′,eµe,i
(k) {exp[i(ωi′,i - ωkF)t] - exp(iωi′,et)
ωe,i - ωkF
+
exp[i(ωi′,i + ωkF)t] - exp(iωi′,et)
ωe,i + ωkF } + c.c. (14)
VLIP(x, t) ) -⟨µ⟩‚E(x, t) (15)
7880 J. Phys. Chem. A, Vol. 108, No. 39, 2004 Nguyen et al.
Because the time of passage of the atoms through the fields’region is much larger than the optical periods of the fields, wecan, in thin samples, ignore the highly oscillatory terms in theabove equations and retain only the slowly oscillating terms.Hence, with the largest contribution coming from thek ) lterms, we have
ForVint(x, t), the interference-induced term, derived from thei * i′ terms, we obtain that
Eliminating the most rapidly oscillating terms we obtain
Retaining only the least oscillatory terms, satisfying theω1,2F
≈ ω2,1 condition of eq 6, we finally have that
The results of eqs 21 and 24 for the non-interference andinterference-induced LIP can be expressed in terms of twopolarizability tensorsø
whereøknon(x) andøint(x) are defined as
and
whereX signifies the outer product between two vectors (e.g.,(a X b)x,y ≡ axby), and where phenomenological widthsΓe havebeen added to the levels to account for spontaneous emissionand other line-broadening mechanisms.
Note that V(2)(x) is the only potential included in thecalculations below sinceµ1,2, and henceV(1)(x, t) is zero forthe choice ofl2 ) l1 + 2, sinceΦ1 and Φ2 are of the sameparity. Note that forΦ1 andΦ2 of opposite parity (e.g.,l2 ) l1+ 1), Vint ) 0, and hence the desired interference term vanishes.The potentialV(2)(x) calculated using eqs 21 and 24 consists ofa series of wells aligned along thex direction, of varying depthsand periodicity. The structure of the potential can be experi-mentally controlled by varying any or all of the following: (a)the frequencies of the two standing wave fields,ω1
F andω2F, (b)
the field strengths ratio,E1(0)/E2
(0), (c) the relative phase betweenthe two standing waves,φF, (d) the relative phase of the initialcoefficients of the superposition state,φM, and/or (e) the ratio|b1/b2|.
III. Computational Methodology
The treatment above provides a general theory for thecoherent control of atomic and molecular trajectories using two
Vnon(x, ω1F, ω2
F) ≈-1
2p∑k)1
2
(Ek(0)Fk(x))2∑
i)1
2
∑neleme
|bi|2[ |µi,e(k)|2
ωe,i + ωkF
+|µi,e
(k)|2
ωe,i - ωkF] (21)
Vint(x, t) )-1
4p∑k)1
2
∑l)1
2
Ek(0)Fk(x)El
(0)Fl(x) ∑neleme
[b1b2/ µ2,e
(l) µe,1(k) ×
{exp[i(ω2,1 - ωkF - ωl
F)t] - exp[i(ω2,e - ωlF)t]
ωe,1 - ωkF
+
exp[i(ω2,1 + ωk,lF )t]- exp[i(ω2,e - ωl
F)t]
ωe,1 + ωkF
+
exp[i(ω2,1 - ωk,lF )t] - exp[i(ω2,e + ωl
F)t]
ωe,1 - ωkF
+
exp[i(ω2,1 + ωkF + ωl
F)t] - exp[i(ω2,e + ωlF)t]
ωe,1 + ωkF } +
b2b1/µ1,e
(l) µe,2(k) ×
{exp[-i(ω2,1 + ωkF + ωl
F)t] - exp[i(ω1,e - ωlF)t]
ωe,2 - ωkF
+
exp[-i(ω2,1 - ωk,lF )t] - exp[i(ω1,e - ωl
F)t]
ωe,2 + ωkF
+
exp[-i(ω2,1 + ωk,lF )t] - exp[i(ω1,e + ωl
F)t]
ωe,2 - ωkF
+
exp[-i(ω2,1 - ωkF - ωl
F)t] - exp[i(ω1,e + ωlF)t]
ωe,2 + ωkF } + c.c.]
(22)
Vint(x, t) ≈ -1
4p∑k)1
2
∑l)1
2
Ek(0)Fk(x)El
(0)Fl(x) ∑neleme
[b1b2/µ2,e
(l) µe,1(k){exp[i(ω2,1 + ωk,l
F )t]
ωe,1 + ωkF
+exp[i(ω2,1 - ωk,l
F )t]
ωe,1 - ωkF } +
b2b1/µ1,e
(l) µe,2(k){exp[-i(ω2,1 - ωk,l
F )t]
ωe,2 + ωkF
+
exp[-i(ω2,1 + ωk,lF )t]
ωe,2 - ωkF } + c.c.] (23)
Vint(x) ≈ -1
2pE1
(0)F1(x)E2(0)F2(x) ∑
neleme
b1b2/{ µ2,e
(1) µe,1(2)
ωe,1 + ω2F
+
µ2,e(2) µe,1
(1)
ωe,1 - ω1F} + c.c. (24)
Vnon(x) ) ∑k)1,2
- Ek(0)ηkøk
non(x) ηkEk(0)
Vint(x) ) -E1(0)η1 øint(x) η2E2
(0) (25)
øknon(x) )
1
2p(Fk(x))2 ∑
i)1
2
∑neleme
|bi|2µi,e X
µe,i[ 1
ωe,i + ωkF - iΓe/2
+1
ωe,i -ωkF - iΓe/2] (26)
øint(x) )1
2pF1(x)F2(x) ∑
neleme
b1b2/[ µ2,e X µe,1
ωe,1 + ω2F - iΓe/2
+
µe,1 X µ2,e
ωe,1 - ω1F - iΓe/2] + b1
/b2[ µe,2 X µ1,e
ωe,1 + ω2F - iΓe/2
+
µ1,e X µe,2
ωe,1 - ω1F - iΓe/2] (27)
Coherent Control in Nanolithography J. Phys. Chem. A, Vol. 108, No. 39, 20047881
standing wave fields. In this section we apply the approachto Rydberg atoms, whose polarizabilities are, conveniently,very large. Additional studies on non-Rydberg atoms areunderway.43
Rydberg atoms, such as those studied in this paper, can betreated as quasi-hydrogenic, with each Rydberg state ap-proximated as a one-electron orbital. Specific details of thepolarizability contributions are given in this section. Rubidiumhas been chosen as a prototype, but the calculations can begeneralized to other alkali metals. We also present details onthe numerical calculation of the deposition patterns.
A. Polarizability Calculation. The Rydberg state of the Rbatom described by principal quantum numbern and angularmomentum quantum numberl has an energy approximated by
whereRy(∞) is the Rydberg constant for the Rb atom calculatedfrom the Rydberg constant assuming infinite mass of the nucleusand me and Mnu are the mass of electron and the Rubidiumnucleus respectively.44,45 Hereδl is the quantum defect whichdescribes deviations of the Rydberg series from atomic hydrogenfor the state with quantum numberl. For Rb, δs ) 3.135,δp ) 2.65, δd ) 1.34 δf ) 0.033;46 states of higherl havenegligible quantum defect and behave like pure hydrogenicstates.
To evaluate the polarizability contributions we first calculatethe transition dipole moments. Applying quantum defect theorygives analytic expressions for the transition dipole moments:44
whereeel is the electronic charge. (Recall that the subscripteindexes the states.) The radial matrix elements⟨nele| r |nl⟩ wereevaluated analytically using the method developed by Kos-telecky and Nieto.47 Note that in evaluating⟨nele| r |nl⟩ thephysical quantum numbersn, l, ne, and le are replaced by thecorresponding effective quantum numbersn*, l*, ne
/, and le/,
which incorporate the proper quantum defect. The quantumdefectsδl andδle in n* ) n - δl andne
/ ) ne - δle representthe effective charge created by the core electrons and the nucleusphenomenologically by shifting the eigenvalues away fromthe hydrogenic values. The effective angular quantumnumbers,l* ) l - δl + I(l) and le
/ ) le - δle + I(le) (I(l), I(le)) 0, 1, 2), on the other hand, do not have physical meaning.They were introduced artificially in ref 47 in order to make theexpression for the evaluation of transition probabilities analyti-cal.
The angular integral⟨leme| cosθ |lm⟩ is simplified by usingthe spherical harmonics
To test the reliability of these expressions we computed thepolarizabilities of an eigenstate|Φi⟩ of Rb:
where i refers ton, l, m and thee sum is overne, le, andme.Specifically, we computed thezzcomponent of the polarizabilitytensor, given by
The computed static polarizabilities (ωF ) 0) for differentRydberg states of Rb are shown in Table 1 along with theexperimental values;48 satisfactory agreement for our purposesis seen. The high polarizability values (both static and dynamic)obtained are useful for lithography insofar as one can userelatively weak electric fields for the controlled deposition. Also,Rydberg atoms can be deflected more easily than their groundstate counterparts. As a result, we can concentrate the atomicRydberg beam by deflecting the Rydberg atoms before thedeposition process, thus reducing the background noise causedby ground state atoms. To this end we note that controlleddeflection of Rydberg molecules, for example, was recentlydemonstrated experimentally by Softley et al.49
Figure 2 depicts the interference contribution to the polariz-ability at parameters given in the figure caption, and Table 2gives, as examples, the real parts of the non-interference andinterference polarizability contributions for several parametersets. The non-interference contribution,øk
non(x), is found to be1 order of magnitude smaller than the interference contribution.Note that the non-interference contribution does not change withφM, and the range of control over the magnitude of thepolarizability is vast.
B. Atomic Density Distribution. Consider, then, a beam ofatoms to be deposited on a surface. The motion of the center-
TABLE 1: Static Polarizabilities (atomic units, i.e., a03) of ns
States of Rb
results
n calcd. exptl.48
8 0.1126696× 106 0.1314194× 106
9 0.3572713× 106 0.4114472× 106
10 0.9496947× 106 0.1078768× 107
11 0.2217497× 107 0.2485978× 107
12 0.4690786× 107 0.5193317× 107
13 0.9180168× 107 0.1004327× 108
14 0.1687154× 108 0.1824944× 108
15 0.2943914× 108 0.3150063× 108
16 0.4918174× 108 0.5208120× 108
17 0.7921468× 108 0.8300864× 108
18 0.1241597× 109 0.1281894× 109
19 0.1954441× 109 0.1925937× 109
øi(ωF) )
1
p∑
e
µi,e X
µe,i[ 1
ωe,i + ωF - iΓe/2+
1
ωe,i - ωF - iΓe/2] (31)
ønlm(zz)(ωF) )
eel2
p∑
neleme
|⟨nele| r |nl⟩|2 4π
3
3(2l + 1)(2le + 1)
4π(-1)2me(1 l le
0 m -me)2 (1 l le0 0 0)2
[ 1
ωe,i + ωF - iΓe/2+
1
ωe,i - ωF - iΓe/2] (32)
En,l ) -Ry(∞)(1 + me/Mnu)
-1
(n - δl)2
(28)
µe,i(z) ) eel⟨nele| r |nl⟩⟨leme| cosθ |lm⟩ (29)
⟨leme| cosθ |lm⟩ ) x4π3
⟨leme| Y10 |lm⟩
) x4π3 (3(2l + 1)(2le + 1)
4π )1/2
(-1)me(1 l le0 m -me
)(1 l le0 0 0) (30)
7882 J. Phys. Chem. A, Vol. 108, No. 39, 2004 Nguyen et al.
of-mass of each atom is governed by Hamilton’s equation ofmotion:
and
wherexi, pi, andM are the position, momentum, and mass ofthe ith atom, respectively. The above equations must be solvedfor the ensemble{xi(t)} of the atomic trajectories, wherexi(t)is thex projection of the position of theith atom, to obtain the
atomic densityF(x, t). Results are determined from a uniforminitial distribution of atoms at timet ) tinter + (Lff /V|), wheretinter is the actual atom-field interaction time andLff is the free-flight distance of the atomic beam.
The computation of the deposition pattern involves thefollowing steps. (1) Att ) 0, a fixed number of atoms isuniformly distributed over a small nozzle segment-aλ2 e x eaλ2, wherea is a constant andλ2 is the longest wavelength ofthe two standing wave fields. This nozzle segment is subdividedinto N - 1 cells of size∆x ) 2aλ2/(N - 1), where∆x is chosensufficiently small so that the force atxi, which is the gradientof the LIP, remains constant over the cell. (2) Given a timet0,the LIP is calculated at every pointxi ) -aλ2 + (i - 1)∆x,(i ) 1,2,....,N). Hamilton’s equations are solved to obtainVi
x )pi
x/M at time t ) t0 + ∆t. The ∆t time step is chosen smallenough to maintain energy conservation. Since there are noforces in the y and z directions, theVi
y and Viz velocity
components are constant. Supplementing these values with theVi
x information obtained from Hamilton’s equations gives theatomic positions (xi, yi, andzi) for each atom in the ensembleat time t ) t0 + ∆t. The procedure is repeated until the atomhits the surface (atzi ) 0). (3) After all the trajectories haveterminated, the density of the atoms on the surface is analyzedby considering a segment of size-bλ2 e x e bλ2, whereb >a. The constantb is chosen large enough to include all depositedtrajectories. Histograms of the data provide the depositeddensity.
IV. Numerical Results and Discussion
The general configuration of the nanofabrication experimentis illustrated in Figure 1. We envision using Rb atoms inRydberg states prepared by an optical excitation from the (52S)ground state.44 After excitation, the Rydberg atoms are mixedwith an inert buffer gas and the mixture is supersonicallyexpanded through a nozzle to narrow down the translationalvelocity distribution.
The average longitudinal velocity of a supersonic source isV| ) x2kγT0/(Mb(γ-1)), wherek is the Boltzmann constant,Mb is the mass of the buffer gas atom,γ ) Cp/Cv is the specificheat ratio of the buffer gas, andT0 is the initial temperature.After exiting the nozzle, the supersonic Rb Rydberg atom beamcan be collimated by letting it pass through a series of slits.50
Alternatively, or additionally, the beam can be collimated bycooling the transverse velocity with a transverse laser field51 toa temperature low enough to ensure no escape from the light-induced potential well. Preparation of the Rb atoms in asingleRydberg level can be done by combining optical pumping and“locking”.52
The LIP experienced by an atom in a single Rydberg statesubjected to a single frequency is periodic, leading to depositionpatterns consisting of equally spaced peaks. A sample is shownin Figure 3, which displays the trajectories and associateddeposition for Rb in the 16s state. As described above, to controlthe deposition pattern we can prepare an initial superpositionof two states using an additional laser pulse.
Coherent modification of the polarizabilities and of therefractive index of atoms and molecules40 can be achieved witheither off-resonant or near-resonant fields. Here, however,because the LIP is governed by a combination of the naturaland altered polarizabilityand the external electric fields, anumber of conditions must be satisfied to make the controlpossible.
Figure 2. Interference contribution to the polarizability (in a.u.) atx) 0, plotted againstφM andφF for thex0.8|16, 0, 0⟩ + x0.2|16, 2, 0⟩superposition state. Two SW fields are of intensityI1 ) 13.3 W/cm2
and I2 ) 132537 W/cm2 at wavelengths ofλ1 ) 41876 nm andλ2 )115781 nm. Upper panel is the real part oføint(x) and the lower one isthe imaginary part.
TABLE 2: Contribution of the Non-interference andInterference Dynamic Polarizabilities (atomic units)a
φF Re(øknon) Re(øk
int)b
0.1282 0.8469× 107 4.6735× 107
2.9380× 107
-0.4525× 107
0.8976 0.5324× 107 2.9380× 107
1.8471× 107
-0.2845× 107
1.6669 -0.8199× 106 -0.4525× 107
-0.2845× 107
0.4381× 106
a Parameters given in the caption of Figure 2.b Results correspondto φM ) 0.1282, 0.8976, 1.6669, respectively.
M∂
∂txi ) pi
x (33)
∂
∂tpi
x ) - ∂
∂xVLIP(x, t)|xi
(34)
Coherent Control in Nanolithography J. Phys. Chem. A, Vol. 108, No. 39, 20047883
Note first, as shown in eq 24, thatVint(x) is proportional tothe product of the field amplitudes,E1
(0) E2(0). By contrast, the
non-interference contribution,Vnon(x), [eq 21] is a sum of termsthat depends on eitherE1
(0)2 or on E2(0)2. Thus, whenE1
(0) , E2(0)
or E1(0) . E2
(0), the periodicity ofVnon(x) dominates, leading toa periodic LIP and to the suppression of coherence effects. Theresulting deposition pattern will thus be periodic. To break thisperiodicity and provide control requires thatE1
(0) ≈ E2(0).
Second, the frequencies of the electric fields have to be near-resonant with an atomic transition. This follows from eqs 26and 27, according to which if the fields are far off-resonancethen we can approximate all theωej - ω2
F, (j ) 1, 2), terms bya common denominator and factor it out of the sums. Further,since in eq 26µje
(k) µej(k), (j ) 1, 2)> 0, whereas in eq 27µ1e
(1) µe2(2)
may take any (complex) value depending onφe, we have that∑neleme µje
(k) µej(k) > ∑neleme µ1e
(1) µe2(2). Consequently, in this case, the
non-interference polarizability contribution would be far biggerthan its interference-induced counterpart, resulting in a periodicLIP and a periodic deposition pattern.
By contrast, if the fields are near-resonant then there existsa Φe which, due to the 1/(ωej - ω2
F) term, dominates the entiresum, which then reduces to one term. In this case thepolarizability depends strongly on the products of transitiondipole matrix elements in the numerators. It is then possible tofind initial states Φ1 and Φ2 such that the interferencecontribution is large enough to significantly alter the depositionpattern.
This qualitative argument has been confirmed by extensivenumerical studies. Among the states studied, we found that the7s+ 7d superposition state is the most suitable, and depositionresults for this superposition state are presented below. In allcases, the field intensities were chosen to beE1
(0) ) E2(0) ) 3.16
W/cm2 and the detuning from the 10p level to be 1 MHz. Thewavelengths of the two fields wereλ1 ) 1.88 µm andλ2 )7.17 µm.
The size of the Rb beam is 15µm, chosen in accord with therequirement-aλ2 e x e aλ2 with a ) 1. The size of thesegment of the substrate where the deposited density iscalculated is 2.5 times larger than the atomic beam size, largeenough to encompass the entire deposition region. Note thatconventional experiments produce atomic beams whose diameterranges from 5 to 100µm. If we were to use a Rb beam that isbigger than 15µm, then the entire deposition pattern would bea periodic repetition of the sub-region pattern formed by the15 µm beam.
Both the system and initial state are chosen to avoid a numberof loss mechanisms. Consider first the issue of spontaneousemission. Rydberg atoms have relatively long radiative lifetimesτ due to their small dipole coupling to the ground state andother low-lying states. For example, in the 7s stateτ ≈ 90 ns.42,53
To minimize spontaneous emission losses, the interaction timeis chosen so thattinter < τ, with the former being controlled bychanging the laser’s waist or the atoms’ longitudinal velocity.Typically (except for the results in Figures 3 and 9),tinter ) 25ns.
A second loss mechanism of concern is ionization, sinceRydberg atoms are easily ionized. Ionization may present aproblem during laser focusing because it alters the opticalpotential felt by the atom, degrading the resolution of thedeposition pattern. Hence, we employ relatively weak focusingfields to avoid ionization. The field necessary to ionize a givenRydberg state isEion ) 1/(16n*4) (a.u.).44 For the superpositionstate considered in this papern* ) 5.67 so that the maximumfield intensity we can allow is 3× 107 V/m. The electric fieldsused below are of maximum intensity 106 V/m, well below thislimit.
Also of concern are stray electric fields. In our coherentcontrol simulations, the smallest electric field is 5× 103 V/m.A typical stray electric field of 10 mV/cm would induce apotential that is 2.5× 107 times smaller than the LIP and wouldnot affect the formation of the deposition patterns. It can,however, modify the composition of the initial 7s+ 7dsuperposition state if weak fields are used to prepare that state.Such a change in the amplitude ratio|b1/b2| will affect thebrightness of the deposition patterns (see the discussion regard-ing Figure 8).
Note that in our coherent control scenario the Rydberg atomsare in a state of relatively low principal quantum number (n )7). In case of high principal quantun numbers (n > 20), thelower limit on the field strength imposed by the ionization willmake the deposition patterns much more sensitive to the strayfields.
Figure 4 shows the total light-induced potential and theinterference-induced and non-interference contributions, as wellas the resultant deposition pattern. As in the periodic case ofFigure 3, the minima of the LIP serve as focusing centers andthe maxima as defocusing centers. Figure 3a shows that thepotential wells associated withVnon(x) are separated by≈ λ2/2.The Vint(x) contribution is, however, aperiodic, containingminima of variable depth and position. In addition, thesepotential wells are steeper that those ofVnon(x), resulting in alarger dipole force. Due to the aperiodic characteristic of theinterference-induced potential, the total LIP is also aperiodic,leading to an aperiodic deposition pattern. Further note that,although the interference-induced contribution contains repulsiveparts, the total LIP is attractive. (Note that the captions providethe parameters used for the computations shown in the figures.)
To more clearly assess the role of the atomic coherence weshow, in Figure 5, the LIP and deposition pattern associated
Figure 3. (Left) Atomic trajectories, with time inµs. Note the darkfocusing regions. (Right) Number of atoms distributed periodicallyalongx (in µm) at tinter ) 0.0012µs. The results correspond to the 16Sstate of Rb atom whereI ) 1.9 × 1013 W/cm2, λ ) 188.5 nm.Deposition plate size) 1319.5 nm and the nozzle width is 565.5 nm.The plate is 0.6µm away from the nozzle.
7884 J. Phys. Chem. A, Vol. 108, No. 39, 2004 Nguyen et al.
with the Vnon(x) contribution to the potential shown in Figure4. The results show a pattern composed of much weaker andbroader peaks, spaced≈ λ2/2 apart, superimposed on asignificant background. Note that the average potential depthof Vnon(x) is ≈ 0.75 meV, which is of the same order ofmagnitude as the total LIP (≈1.0 meV). Thus the broadeningof the deposited peaks is not caused by the small decrease inthe depth of the minima, but rather due to a change in the shapeof the LIP. Thus, in addition to introducing aperiodicities inthe deposition patterns,Vint plays an important role in sharpeningthe deposition pattern. Specifically, a comparison of the peaksin Figures 5 and 4 show that the introduction of the interferencecontribution narrows the peaks by a factor of 3 and increasesthe contrast with the background by a factor of 5.
The coherent control approach provides for control througha wide variety of experimentally adjustable control parameters.Consider firstφM, the relative phase betweenb1 and b2. Thisterm affectsVint(x), which depends onφM through b1b2
/ andb1/b2 [eq 27]. The phaseφM alters relative contributions of the
interference-induced and non-interference contributions, result-ing in different total LIPs and different deposition patterns. Thisis demonstrated numerically in Figure 6, where the depositionpatterns for a fixedφF ) (π/3), and variableφM ) 0 in (b),(π/3) in (d), andπ in (f) are shown. The number of depositedpeaks varies from six in panel (b) to four in panel (d) and ninein (f). It is evident that the positions, number, and the heightsof the peaks are very sensitive to changes inφM. For examplein panel (b), in the central region of the plate (from-3 to 3µm) the deposition peaks are separated by≈ λ1, whereas theyare separated byλ2/2 in the same region in panel (d). Because
φF is fixed, the Vnon(x) contribution to LIP is unchanged.Therefore, the variation in the total LIPs shown in (a), (c), and(e) results from the changes inVint(x).
A second control parameter isφF, the relative phase betweenthe two laser fields. Figure 7 shows the deposition patterns fora fixedφM ) 0, atφF ) 0 in (b), (π/3) in (d), andπ in (f). Weobserve from the figure that there are five significant peaks in(b), six in (d), and six in (f). As compared toφM, the number ofthe peaks depends less sensitively onφF, but changingφF allowsone to manipulate the location of the minima and maxima, i.e.,the focusing and defocusing centers, of the total LIP, as shownin panels (a), (c), and (e).
In addition toφM, the initial superposition state depends onthe amplitude ratio|b1/b2| which can be altered experimentallyby changing the pulse parameters in the preparation step of thesuperposition state. Figure 8 shows the dependence of the
Figure 4. (a) Contributions to the light-induced potential:Vnon(x)(dotted line) andVint(x) (full line); (b) the total light-induced potentialV(x) ) Vnon(x) + Vint(x); and (c) the deposition pattern for thex0.8| 7, 0, 0⟩ + x0.2|7, 2, 0⟩ superposition state. The size of thebeam and plate is 15µm. A total of 1000 atomic trajectories have beenused, with the transverse velocity taken as zero. The field intensity is3.16 W/cm2. Other parameters are:φM ) 0, φF ) (π/3), tinter ) 0.025µs, λ1 ) 1.88µm, andλ2 ) 7.17µm. Here and below all angles are inradians.
Figure 5. Same as in Figure 4. (a) The noninterference potential,Vnon(x), and (b) the resulting deposition pattern.
Figure 6. Panels (a), (c), and (e): LIP and deposition patterns (panelsb, d, f) associated with thex0.8| 7, 0, 0⟩ + x0.2|7, 2, 0⟩ superposi-tion state.φF ) (π/3) andφM ) 0 in (a,b), (π/3) in (c,d), andπ in (e,f).
Coherent Control in Nanolithography J. Phys. Chem. A, Vol. 108, No. 39, 20047885
deposition on|b2|, where|b2|2 < 0.2, consistent with preparationvia perturbation theory. Note that changing|b2| does not affectthe location of the peaks. It does, however, modify the heightof the peaks, altering the brightness of the deposited pattern.
In addition to these control parameters arising from thecoherent control scenario, there are other traditional controlparameters that originate from the particular experimental setupin nanolithography. Specifically one can altertinter, the time ofinteraction of the atoms with the light source, andLff , the free-flight distance during which the atoms travel after havinginteracted with the laser and before colliding with the surface.Because of the deleterious effects of the transverse velocity,increasing the free-flight distance generally degrades the sharp-ness of the deposition pattern; in our caseLff ) 0. Similarly,tinter, the interaction time (which must be less than the lifetimeof the Rydberg state), has a direct effect on the resolution ofthe deposition pattern. Figure 9 shows the result for differentinteraction times. Note that whentinter is small (e.g., Figure 9a),the atoms do not have enough time to be influenced by the LIP,
and focusing cannot be achieved. On the other hand iftinter istoo long, the optical force deflects the atoms more than it should,causing broadening and splitting of the peaks (Figure 9c andd). Among the examples shown, the best interaction time is0.025 µs (Figure 9b). Note that in case of atoms in a singlestate interacting with one laser field, it is possible to derive ananalytic expression giving the optimaltinter that gives the bestfocusing.54 In our case, however, the atoms are prepared in asuperposition state, and the total LIP consists of potential wellsof various depth with aperiodic spacing. One cannot thereforedetermine the optimaltinter analytically. However, it can beobtained by using an optimal control scenario.
Our coherent control results have assumed a perfectlycollimated atomic beam. To account for a less-than-idealsituation requires that we include the transverse velocitydistribution. Theoretically, the transverse kinetic energy of theatom must be smaller than the depth of the potential well inorder to ensure no escape from the well. This condition gives
the maximum allowed transverse velocityV⊥max ) x2|VLIP
max|/M.This V⊥
max is a function of the atom eigenstate, the lightintensity, and frequency. In our examples, the average depth ofthe potential is∼ 0.5 meV, from which a maximum allowedtransverse velocity of 34 m/s can be derived. However, computersimulation has shown that even a small transverse velocity ofabout 5 m/s can broaden the peaks considerably. The broadeningis due to the fact that atoms with transverse velocities can gainbigger transverse momentum and hence leave the focus. Oneway to counter this problem is to optimize the control parametersso as to reduce the damage caused by the transverse velocity.57
Chromatic aberration resulting from the longitudinal velocityspread has less degrading effect on the line width. Indeed,including a velocity spread as large asδV|/V| ≈ 1 would broadenthe line width by 36% with respect to that obtained with amonochromatic beam.58
In our numerical simulations, the atoms were treated as point-like particles and their center-of-mass motion calculated ac-cording to the laws of classical mechanics. In reality, the size
Figure 7. LIP (panels a,c,e) and deposition patterns (panels b,d,f)associated with thex0.8| 7, 0, 0⟩ + x0.2|7, 2, 0⟩ superposition state.φM ) 0 andφF ) 0 in (a,b), (π/3) in (c,d), andπ in (e,f).
Figure 8. Deposition patterns associated with theb1|7, 0, 0⟩ + b2|7,2, 0⟩ superposition state.|b2|2 ) 0.05 in panel (a), 0.1 in (b), 0.15 in(c), and 0.2 in (d). Other parameters areφM ) 0, φF ) (π/3).
Figure 9. Deposition patterns associated with thex0.8|7, 0, 0⟩ +x0.2|7, 2, 0⟩ superposition state.tinter ) 0.015µs in (a), 0.025µs in(b), 0.035µs in (c), and 0.045µs in (d). Other parameters areφM ) 0,φF ) (π/3).
7886 J. Phys. Chem. A, Vol. 108, No. 39, 2004 Nguyen et al.
of the Rydberg atom, which is equal ton2a0, must be takeninto account. Thus, the 10s state is 5.9 nm, whereas that of a100s state atom is 5900 nm. The atom’s unusually large sizewould decrease the resolution of the deposition were it to remainin the Rydberg state after striking the surface. Fortunately, theweakly bound Rydberg electron undergoes tunneling ioniza-tion55,56 as the atom approaches to within≈3.7n2a0 of thesurface. The remaining ion is then neutralized on the surfaceby an Auger process. Hence we expect that the ionizationprocess will reduce the size of the Rydberg atom to that of theground-state atom, preventing the degradation of the resolutionof the deposited line. In our model the distance between thenozzle and the plate is 12.5µm, and the distance at which theionization occurs is≈0.04µm. Hence tunneling ionization doesnot degrade the deposition pattern. However, once the atomsare deposited on the surface, growth and diffusion phenomenacan broaden the line width by 20-30 nm and reduce the contrastto ≈1:1.59 One solution is to optimize the experimentalparameters in the standing wave fields and the atomic beam. Acontrol scenario for surface diffusion would also be useful.
Finally, we note that despite the improvement in line widthdue to the introduction of the interference contribution, theresultant lines are still relatively broad, with widths on the orderof 50 nm. This width results from the fact that we have chosento focus on producing aperiodic sets of lines via a scenario thatallows for control over the line pattern by manipulating the laserparameters, and to do so with Rydberg atoms that allow forweak fields, using a fixed distance between the source and thesurface. No explicit focus was placed on narrowing thedeposition lines. However, as the discussion in this paper madeclear, there are a wide variety of system parameters that can bevaried in order to target alternative features, such as narrowdeposition lines. Under these circumstances, given the largenumber of possible system parameters, the best approach is todesign an optimization scheme to reach specific targets. Thisis discussed in a future paper,57 where we demonstrate that, forexample, one can optimize to produce a single Rb peak asnarrow as one nanometer in width.
V. Summary
In this paper we have shown that bichromatic control ofRydberg atoms in superposition states provides a means ofproducing controllable aperiodic deposition patterns on surfacesthat can be manipulated by changing the constitution of theinitial superposition state and the phase between the two incidentlight fields. Other control parameters, such as the interactiontime, were also shown to modify the resulting pattern. Althoughthe creation of a completely arbitrary pattern is still in the future,it is clear that coherent control offers useful flexibility infabricating nanoscale patterns. Explorations of producing arbi-trary patterns using multicolored optimal and coherent controlmethodologies are underway.57
Acknowledgment. PB and MS acknowledge the longstand-ing personal support, interest, and contributions of ProfessorRichard Bersohn to their lives. Richard was a shining exampleof personal integrity and scientific commitment. He will besorely missed. We thank the U.S. Office of Naval Research forsupport and H. R. Sadeghpour (Harvard) for suggesting thatwe treat Rydberg atoms.
References and Notes
(1) For a review see, Ito, T.; Okazaki, S.Nature2000, 406, 1027.(2) Bell, A. S.; Pfau, T.; Drodofsky, U.; Stuhler, J.; Schulze, T.;
(3) Lison, F.; Adams, H. J.; Haubrich, D.; Kreis, M.; Nowak, S.;Meschede, D.Appl. Phys. B1997, 65, 419.
(4) Berggren, K. K.; Younkin, R.; Cheung, E.; Prentiss, M.; Black, A.J.; Whitesides, G. M.; Ralph, D. C.; Black, C. T.; Tinkham, M.AdV. Mater.1997, 9, 52.
(5) Thywissen, J. H.; Johnson, K. S.; Younkin, R.; Dekker, N. H.;Berggren, K. K.; Chu, A. P.; Prentiss, M.; Lee, S. A.J. Vac. Sci. Technol.1997, B15, 2093.
B54, 375.(13) Prentiss, M.; Timp, G.; Bigelow, N.; Behringer, R. E.; Cunningham,
J. E.Appl. Phys. Lett.1992, 60, 1027.(14) McClelland, J. J.; Scholten, R. E.; Palm, E. C.; Celotta, R. J.Science
1993, 262, 877.(15) McGowan, R. W.; Giltner, D. M.; Lee, S. A.Opt. Lett.1995, 20,
2535.(16) Cook, R. J.; Hill, R. K.Opt. Commun.1982, 43, 258.(17) Balykin, V. I.; Letokhov, V. S.; Ovchinnikov, B. Yu.; Sidorov, A.
I. Phys. ReV. Lett. 1988, 60, 2137.(18) Aminoff, C. G.; Steane, A. M.; Bouyer, P.; Desbiolles, P.; Dalibard,
J.; Cohen-Tannoudji, C.Phys. ReV. Lett. 1993, 71, 3083.(19) Kasevich, M. A.; Weiss, D. S.; Chu, S.Opt. Lett.1990, 15, 607.(20) Kasevich, M. A.; Chu, S.Phys. ReV. Lett. 1991, 67, 181.(21) Gould, P. L.; Ruff, G. A.; Pritchard, D. E.Phys. ReV. Lett. 1986,
56, 827.(22) Lawall, J.; Prentiss, M.Phys. ReV. Lett. 1994, 72, 993.(23) Renn, M. J.; Montgomery, D.; Vdovin, O.; Anderson, D. Z.;
Wieman, C. E.; Cornell, E. A.Phys. ReV. Lett. 1995, 75, 3253.(24) Ito, H.; Nakata, T.; Sakaki, K.; Ohtsu, M.; Lee, K. I.; Jhe, W.
Phys. ReV. Lett. 1996, 76, 4500.(25) Metcalf, H.; van der Straten, P.Phys. Rep.1994, 244, 203.(26) Askar’yan, G. A.SoV. Phys. JETP1962, 15, 1088.(27) Letokhov, V. S.JETP Lett.1968, 7, 272.(28) Ashkin, A.Phys. ReV. Lett. 1970, 24, 156.(29) Ashkin, A.Phys. ReV. Lett. 1978, 40, 729.(30) Chu, S.; Bjorkholm, J. E.; Ashkin, A.; Cable, A.Phys. ReV. Lett.
1986, 57, 314.(31) Adams, C. S.; Sigel, M.; Mlynek, J.Phys. Rep.1994, 240, 143.(32) Chu, S.ReV. Mod. Phys.1998, 70, 686.(33) Seideman, T.J. Chem. Phys.1997, 106, 2881.(34) Seideman, T.Phys. ReV. A 1997, 56, R17.(35) Gordon, R. J.; Zhu, L.; Schroeder, W. A.; Seideman, T.J. Appl.
177. Shapiro, M.; Brumer, P.AdV. Atom. Mol. Opt. Phys.2000, 42, 287.(38) Rice, S. A.; Zhao, M.Optical Control of Molecular Dynamics;
John Wiley & Sons: New York, 2000.(39) Shapiro, M.; Brumer, P.Principles of the Quantum Control of
Molecular Processes; John Wiley & Sons: New York, 2003.(40) McCullough, E.; Shapiro, M.; Brumer, P.Phys. ReV. A 2000, 61,
041801(R).(41) Dey, B.; Shapiro, M.; Brumer, P.Phys. ReV. Lett.2000, 85, 3125.(42) Gounand, F.J. Phys. (Paris)1979, 40, 457.(43) Nguyen, N. A.; Shapiro, M.; Brumer, P., work in progress.(44) Gallagher, T. F.Rydberg Atoms; Cambridge University Press:
(46) Lebedev, V. S.Physics of Highly Excited Atoms and Ions; Springer-Verlag: New York, 1998.
(47) Kostelecky, V. A.; Nieto, M. M.Phys. ReV. A 1985, 32, 3243.(48) O’Sullivan, M. S.; Stoicheff, B. P.Phys. ReV. A 1985, 31, 2718.(49) Procter, S. R.; Yamakita, Y.; Merkt, F.; Softley, T. P.Chem. Phys.
Lett. 2003, 374, 667.(50) Ramsey, N.Molecular Beams; Clarendon: Oxford, 1956.(51) Sheehy, B.; Shang, S.-Q.; van der Straten, P.; Metcalf, H.Chem.
Phys.1990, 145, 317.
Coherent Control in Nanolithography J. Phys. Chem. A, Vol. 108, No. 39, 20047887
(52) Gould, P. L.; Ruff, G. A.; Martin, P. J.; Pritchard, D. E.Phys. ReV.A 1987, 36, 1478.
(53) Theodosiou, C. E.Phys. ReV. A 1984, 30, 2881.(54) Berggren, K. K.; Prentiss, M.; Timp, G. L.; Behringer, R. E.J.
Opt. Soc. Am. B1994, 11, 1166.(55) Burgdorfer, J.; Lerner, P.; Meyer, F. W.Phys. ReV. A 1991, 44, 5674.
(56) Hill, S. B.; Haich, C. B.; Zhou, Z.; Nordlander, P.; Dunning, F. B.Phys. ReV. Lett. 2000, 85, 5444.
(57) Nguyen, N. A.; Shapiro, M.; Brumer, P., submitted.(58) McClelland, J. J.J. Opt. Soc. Am. B1995, 12, 1761.(59) Behringer, R. E.; Natarajan, V.; Timp, G.Appl. Surf. Sci.1996,
104, 291.
7888 J. Phys. Chem. A, Vol. 108, No. 39, 2004 Nguyen et al.
